"Topping trees" is a term used in arboriculture. It means cutting older trees down almost to the top of the trunk. Topping trees is sometimes used as a less expensive alternative to their full removal, which can be quite costly in the case of large old trees. It is done not with an eye to what is best for the tree, but because the homeowner is in a pinch. Some types of trees may die after topping, but this fact often holds little importance to the homeowner, who may actually be glad to have the tree die if it is unwanted.
A similar term -- with a longer tradition behind it -- is "pollarding." But there is a difference between topping trees and pollarding them. More thought goes in to pollarding. For example, it is known that only certain types of trees respond well to the practice, so pollarding is limited to those trees. In contrast with topping, pollarding is first performed when a tree is young. Pollarding is also an ongoing operation, while topping is often done just once, as a stop-gap measure.
When hiring arborists or hiring tree services to trim tree limbs that hang threateningly over your home, make sure they have no intentions of topping your tree -- if the tree has any importance to you. Topping, in such circumstances, is a highly disreputable procedure.